They are indeed right next door. Also, it’s a crowded universe, and no-one wants to be noticed, so just showing up at Tau Ceti or whatever would be very unwise; there might already be someone there.

The major bit of artistic license is that their Alpha Centauri is way more broken than ours; the real one isn’t all that badly behaved.

It’s implied that the whole situation is very unusual; interstellar invasions don’t happen as a rule and they’re only doing it because their star(s) is basically broken.